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<p>
Currently in the LMFDB, we have data on L-functions associated to
{{ KNOWL("mf.elliptic",title="holomorphic cusp forms")}} on Hecke congruence groups
$\Gamma_0(N)$ with character $\chi$.  
These L-functions \(L(s,f) = \sum a_n n^{-s} \) have an 
{{ KNOWL('lfunction.euler_product', title='Euler product')}}
of the form
\[
   L(s,f)= \prod_{p\mid N}  \left(1 - a_p p^{-s} \right)^{-1}\prod_{p\nmid N}  \left(1 - a_p p^{-s} + \chi(p) p^{-2s} \right)^{-1}
\]
and satisfies the 
{{ KNOWL("lfunction.functional_equation",title="functional equation")}} of the form 
\begin{equation} 
\Lambda(s,f) = N^{s/2} \Gamma_{\mathbb{C} }
\left(s + \frac{k-1}{2} \right)\cdot L(s, f) =
\varepsilon \Lambda(1-s,f),
\end{equation}
where  $N$ is the 
{{ KNOWL("mf.elliptic.level",title="level")}}, $k$ is the
{{ KNOWL("mf.elliptic.weight",title="weight")}}
and \(a_n n^{\frac{k-1}{2} } \) 
are algebraic integers. 
</p>

<p>
In the plot below, L-functions are organized by the ordered pair \((N,k)\).
For a given \((N,k)\), the color indicates the {{ KNOWL('lfunction.sign', title='sign')}}
of the functional equation, and the horizontal 
grouping indicates the {{ KNOWL('nf.degree', title='degree')}} of the field containing the arithmetically normalized coefficients. 

See the {{ KNOWL('lfunction.degree2holo.key', title='legend') }} for more details.
</p>
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